1. Key idea: differential thermal contraction

   In a bimetallic strip, the two metals are firmly joined together. If temperature changes, each metal would like to change its length by $\Delta L = \alpha L \Delta T$ where $\alpha$ is the coefficient of linear expansion.

2. Given information

   • Metal $A$ has a higher coefficient of expansion than metal $B$.
   • The strip is placed in a cold bath, so temperature decreases.
   • Therefore, both metals try to contract, but metal $A$ tries to contract more than metal $B$ because $\alpha_A > \alpha_B$

3. How bending occurs

   Since the two strips are attached together, they cannot contract by different amounts independently.

   • The metal that wants to contract more becomes effectively the inner side of the bend.
   • The metal that contracts less becomes the outer side.

   So on cooling, the strip bends towards metal $A$.

4. Direction of bending from the figure

   From the standard mounting shown in such problems, metal $A$ is on the left side of the strip arrangement. Hence, on cooling, the strip bends towards the left.

5. Checking options

   • A: Neither bend nor shrink — false, because cooling causes contraction and differential contraction causes bending.
   • B: Bend towards the left — correct.
   • C: Not bend but shrink — false, because unequal contraction causes bending.
   • D: Bend towards the right — false.

6. Final answer

   $\boxed{\text{Bend towards the left}}$

Therefore, the correct option is B.